Hypergeometric Series, a Barnes‐Type Lemma, and Whittaker Functions

Central to the Fourier development of automorphic forms on GL(n, R) are the class 1 principal series Whittaker functions W n,a ( z ), which were first studied systematically by Jacquet [ 13 ]. (See Section 2 below for the definition of W n,a ( z ).) Of particular interest are the Mellin transforms M n,a ( s ) ( s ∈ C n −1 ) of W n,a ( z ). (See equation (2.11) below.) For example, such transforms, and analogous Mellin transforms of products of Whittaker functions, arise as archimedean Euler factors for certain automorphic L ‐functions (see [ 5 , 6 , 8 , 21 , 22 ] for discussions and examples.) Moreover, M n,a ( s ) has relevance to the problem of special values of Whittaker functions; cf. [ 7 ] in the case n = 3. Friedberg and Goldfeld [ 11 ] have shown M n,a ( s ), for general n , to have analytic continuation and to satisfy certain recurrence relations. However, explicit formulae for these transforms have, until the present work, been deduced only for n ⩽ 4. In particular, both M 2, a ( s ) (cf. [ 4 , 25 ]) and M 3, a ( s ) (cf. [ 7 , 9 , 19 ]) are expressible as (some powers of 2 and π times) ratios of gamma functions. On the other hand, M 4, a ( s ) (cf. [ 22 ]) may be realized essentially as a hypergeometric series of type 7 F 6 (1), or equivalently as a sum of two series of type 4 F 3 (1). (See Section 2 below for a brief general discussion of hypergeometric series.)